Huitiemes Entretiens du Centre Jacques Cartier URL:http://www.emath.fr/proc/Vol.2/
SOME EXTENSIONS OF APPROXIMATE
CONTROLLABILITY RESULTS TO INVERSE PROBLEMS
AXEL OSSES AND JEAN-PIERRE PUEL
Abstract. We study here two inverse problems for the Laplace
equa-tion and for a generalized Stokes system, using approximate controllabil-ity methods. This enables us to give constructive results. The method is then used to develop numerical algorithms. Numerical results are given for the case of Laplace equation.
Key words: Laplace Equation, Stokes System, Inverse Problems,
Approx-imate Controllability, Duality, Minimization Problem.
1. Introduction
We present here some extensions of a method which has been developped in 3] for the study of approximate controllability in semilinear heat equa-tions. We consider here some elliptic problems, namely the Laplace equation or a generalized Stokes system for which, for example, the boundary value is unknown on a part of the boundary but some \measurements" of the solution are given on an internal surface. The problem is then to retrieve (an approximation of) the boundary value from the given measurements. This can be viewed as an inverse problem for which approximate controlla-bility techniques apply and give a constructive method which can be used for numerical algorithms and eective computations.
In section 2 we consider the case of the Laplace equation and we give a precise description of the method.
In section 3, we present the numerical experiments and results while in section 4 we consider the case of a generalized Stokes system for which numerical calculations are in progress.
Our results make an extensive use of unique continuation properties which are classical for the case of Laplace equations but rely on new results by C. Fabre and G. Lebeau (1],2]) in the case of Stokes systems.
2. Inverse problem for the Laplace equation
Let be a bounded regular open set of<
N. We denote by ; the boundary
of , and by the unit exterior normal vector at a point of ;. We suppose
that ; = ;0
;
1, where ;0 and ;1 are nonempty and relatively open.
For any v2L 2(;
0) we know that there exists a unique solution
y=y(v)
of
8 <
:
;y=f in ,
y=v on ;
0,
y= 0 on ;
1,
(1) where f is a given function in L
2().
Remark 1. Problem (1) is solved by a transposition method (see 7]) and
we only obtain y 2L
2() and y 2L
2(). Nevertheless, it is classical to
show that if U is a neighborhood of ; 0,
y is regular in ;U.
S
Ω
Γ0 ν
Figure 1. Domain and curveS.
We are now given an internal regular surface S (of codimension 1) such
that S . From Remark 1, the trace ofy(v) on S has a perfect meaning
in H 1=2(
S). Given y 1
2 L 2(
S) and > 0, our problem is now to look for v2L
2(;
0) such that
jy(v) =S
;y 1
j L
2 (S)
(2)
Remark 2. 1)y
1represents some \measurements" of y(v)
=Sand
v=y(v) =;
0
is \unknown". We then try to nd v such that y(v)
=S is \almost" equal to
the measurements.
2) Of course = 0 would be optimal but the problem is then in general
impossible as can be seen by regularity reasons for example. Take f = 0
then y(v) is analytic in and y
1 is only taken in L
2( S).
3) Our approach is inspired by approximate controllability methods as our problem is equivalent to showing that the set fy(v)
=S
v 2 L 2(;
0) g is
dense in L 2(
S).
Another classical approach (corresponding to an optimal control philoso-phy) for solving the inverse problem is to minimize
H(v) = 12 Z
S jy(v)
S ;y
1 j
2 ds+
Z
; 0
jvj 2
d , >0
The two methods present some similarities but are dierent. We will not consider the second one here.
In order to state our results, we have to make a geometrical hypothesis on S. Let us dene
C=fc2C(01] )8t2]01c(t)2;S c(0)2Sc(1)2; 0
g
The geometrical condition on S can be stated as follows
S 2 and8x2S9c x
2Cc x(0) =
x: (3)
Let us now dene
S ext=
fx2;S 9c2C9t2]01c(t) =xg
S
int= ;S
ext
;00 =
fx2; 0
9c2Cc(1) =xg:
We can easily prove the following properties
Proposition 3. (i) ;00 is a nonempty open subset of ;0 and without loss
of generality we can assume that ;0= ;00 (by taking
v= 0 on ; 0
(b)
Γ0 Γ
(e)
0
Γ0
(c)
S
S S
Γ0
(d)
Γ
Γ0 0
S S
(a)
Figure 2. In cases (a), (b),(c), (3) is satised and in cases
(d), (e), (3) is not satised.
(ii) S
ext is a nonempty open set and we have :
S @S
ext ;
00
@S
ext.
Moreover, if S
ext has several connected components, the boundary of each of
them contains a nonempty open subset of ;00.
(iii) S
int may be empty but if not, @S
int
S;.
The following gure shows cases where (3) is satised and cases where (3) is not satised.
Theorem 4. Let us assume that S satises condition (3) . Then for every
y 1
2L 2(
S) and any >0, there existsv2L 2(;
0) such that jy(v)
=S ;y
1 j
L 2
(S)
:
Moreover, if
;y=f in
y= 0 on ;
we can takev=; @
^ @
=;
0 where (denoting by
S the Dirac mass on the surface S)
;^= ^ 0
: S in
^
=;= 0
and ^
0 minimizes over L
2(
S) the functional
J( 0) = 12
Z
; 0
j @ @ j
2
d+j
0 j
L 2
(S) ;
Z
S
(y 1
;y =S)
0
ds: (4)
with
;=
0 :
S in
=;= 0
(5) Proof. It can be shown directly, using Hahn Banach theorem, thatfy(v)
=S v 2 L
2(; 0)
g is dense in L 2(
S) but we will give a proof using a duality
Take 0
2L 2(
S) and J(
0) dened by (4). Notice that after translation
by y we can assume thatf = 0. It is easy to show thatJ is strictly convex
and continuous on L 2(
S). Moreover one can show the following
Lemma 5. The functional J is coercive if and only if the following unique
continuation property holds.
8 < : ;= 0 : S in
= 0 on ;
@
@ = 0 on ;
0
(6) implies
= 0 in and
0 = 0 (7)
Let us now show that whenS satises condition (3), the unique
continu-ation property holds true. We have from (6)
8 <
:
;= 0 inS ext
= 0 on ;
00 @
@ = 0 on ;
00
Then, by classical unique continuation property for elliptic equations (see for example 11] for a general case and the bibliography therein), = 0 in S
ext. Then
= 0 on@S
ext and therefore
= 0 onS. Now we have
;= 0 in S
int
=@Sint = 0
so that = 0 on S
int, and therefore
= 0 in and
0 = 0, which shows
the unique continuation property.
From the previous lemma we know that J is coercive. Then J has a
unique minimum ^
0. It is easy to show that ^
0
6
= 0, jy 1 ;y =S j L 2 (S)
>,
the casejy 1 ;y =S j L 2 (S)
being trivial asv= 0 is solution to our problem.
In the case jy 1 ;y =S j L 2 (S)
>, we obtain
(J 0( ^
0)
0) = 0
8
0 2L
2( S)
so that, if denotes the solution of (5) associated with
0, we have
8 0 2L 2( S) Z ; 0 @^ @ @ @ d+ Z S ^ 0 j^
0 j L 2 (S) : 0 ds; Z S (y 1 ;y =S) 0
ds= 0
Take ^v=; @
^ @=;
0 and ^
y=y(^v). we obtain Z ; 0 @^ @ @ @ d= Z S (^y =S ;y =S) 0 ds: Therefore, 8 0 2L 2( S) Z S (^y =S ;y 1) 0
ds=; Z
S
^
0 j^
0 j L 2 (S) : 0 ds
which shows that
^ y =S ;y 1 = ; ^ 0 j^
0 j L 2 (S) and
h
Γ0h
Sh
Ω
Figure 3.
Remark 6. 1) As it was shown in 4, 5, 6], by a duality argument, one can
show that ^v=; @
^ @
=;0 is such that jvj^
L 2
(; 0
) =
Minfjvj L
2 (;
0 )
jy(v) =S
;y 1
j L
2 (S)
g:
By considering a dierent functional J as in 3], we could nd a control ~v
minimizing the L
1 norm among admissible controls.
2) We can extend the method to a general second order elliptic operator such that the adjoint operator is coercive and regular.
3) Using a xed point argument, we can extend the result to a semilinear problem of the form
8 <
:
;y+g(y) =f in ,
y=v on ;
0,
y= 0 on ;
1,
(8)
looking for v 2 L 2(;
0) such that jy(v)
=S ;y
1 j
L 2
(S)
, when g is a non
decreasing Lipschitz function. If we take g(y) =jyj p;1
y withp>1, we can
show that the result of approximate controllability is no more valid. 4) The detailed proofs and various extensions will be given in 8].
3. Numerical Method and Experiments
We take here for a polygonal domain of < 2, for
S a polygonal internal
curve in and f = 0. We consider a triangulationT
h of such that S and
;0 are union of triangles of T
h, as in Figure 3 below.
We suppose that ;0 consists of (
Q;1) edges,S consists of (M;1) edges
and N is the total number of interior vertices in . Let
V 0h=
f2C
0()
8T 2T h
=T 2P
1
=;= 0 g
and let w 1
:::w
N be a basis of V
0h. We denote by L
h a nite dimensional
subspace of L 2(
S) consisting of piecewise constant functions on each edge
of a triangle T 2 T
h contained in
S (we could also take piecewise linear
functions), and by 1
:::
M a basis of L
h.
If 2L
h, we call T
h the solution of
T h
2V 0h
R
rT
h
rdx=
R S
ds 82V
0h :
We consider the matrices
G k l=
Z ; 0 @T h k @ @T h l @ d C k l=
Z S k l ds f 1k = Z S y 1 k ds:
If 2L
h we have = P M k =1 k
k and if
t= (
1 :::
M), we consider the
functional
J h(
) = 12 t G; t f 1+ ( t C) 1=2 (10)
and the minimization problem
J h(^
) = min 2<
M J
h(
): (11)
Then, if ^h= P M k =1 ^ k
k, the approximate control is given by
^
v h =
; @T
h^h @ =; M X k =1 ^ k @T h k @ :
The numerical problem consists in solving M problems (9) with right hand
sides
k and then in minimizing J
h.
Remark 7. 1) The problem of computing the normal derivatives is not
simple and may generate large errors. Various choices can be made for this computation.
2) Matrix G is badly conditionned so we use a preconditionner = diag(G
;1=2 ii )
0 and we work with double precision variables.
The following computations have been made using MODULEF for the mesh generation and elementary matrices, a Cholesky decomposition for solving the NN linear systems (withM right hand sides) and the BFGS
method (see 10]) for the minimization problem.
In Figure 4 we present three numerical experiments corresponding to dif-ferent positions of ;0,
S and dierent . For all these three cases we have
taken y
1 = 1 on
S. The control ^v
h is schematically represented normally
to ;0. The value of the solution ^ y
h is represented by levels of grey and the
white region near each curve S corresponds to the level 1:000:03.
On Figure 5 we consider two other functions y
1 in the geometry of rst
case in Figure 4 :a sine function and a Heaviside function. There is an important error for the case of Heaviside function and this was expected as the solution ^y must be very regular onS.
Figure 6 shows the relation between the relative theoretical and numerical errors and in the rst example of Figure 4 (the results for all examples
are similar), where the relative theoretical error is
jy^;y 1 j L 2 (S) jy 1 j L 2 (S) = ( jy 1 j L 2 (S )
if <jy 1
j L
2 (S)
1 if jy
1 j
L 2
(S)
and the relative numerical error is
e() = jy^
0
Γ
S
0
Γ
S S
Γ0
Figure 4.
These results show that a mesh renement does not improve the precision below a critical value of . We don't have a precise explanation of this
phenomenon at the moment.
4. Inverse Problem for a Generalized Stokes System
We keep the same notations as in Section 2 and we consider the standard functional spaces
H=fz 2(L 2())N
divz= 0g
V =fy2(H 1 0())
N
divy= 0g
L 2 0(;
0) =
fv 2(L 2(;
0)) N
Z
;0
Nodes on the curve S
Values on the curve S
10 20 30
5 15 25
-1 0 1
-0.5 0.5
calculated value desired value
Values on the curve S
Nodes on the curve S
Values on the curve S
0 5 10 15 20 25 30
-1 0 1
-0.5 0.5
calculated value desired value
Values on the curve S
Figure 5.
relative alpha
relative error
-3
10 10-2 10-1 100 101 102 103
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
initial mesh refinement 50% refinement 100% theoretical Error on S
numerical
theoretical
Let us take a2(L ()) diva= 0 and f 2(L ()) . Forv2L 0(;
0) we
consider the following problem
8 > > < > > :
;y i+ P N j=1 @ @x j( a j y
j) = f
i ;
@p @x
i in ,
i= 1:::N divy= 0
y =; 0 = v y =; 1 = 0
:
(12)
Proposition 8. There exists a unique solution (yp) = (y(v)p(v)) to (12)
with y 2 H, p 2 L 2 loc()
=<. The correct denition of
y is given by the
transposition method as follows : for every h2(L
2())N, let (
q)2V L 2()
=< be solution of 8
<
:
;;(a:r)=h;rq in , div= 0
=;= 0
:
(13)
Then (q)2(H 2())N
H
1() (see 12]) and we have
8h2(L 2())N
Z y:hdx= Z f:dx; Z ; 0 v:( @ @
;q)d (14)
Remark 9. Condition diva = 0 is only required for having an existence
result in (13)
The result of Proposition 8 comes from the classical application of the transposition method. Interpretation of (14) gives (12) except for the bound-ary condition which does not make full sense for y 2 H. We completely
recover the normal component of the boundary condition whereas the tan-gential components are obtained only if y is more regular. Anyway, there
exists a unique solution to (14) and we can show that if U is a
neighbor-hood of ;0, then
y2(H 1(
;U))
N so that if
S is an internal surface such
that S , the trace y
=S makes perfect sense. Our problem is then the
following :Given y 1
2(L 2(
S)) N and
>0, nd v2L 2 0(;
0) such that jy(v)
=S ;y 1 j (L 2 (S))
N :
Theorem 10. If S satises the geometrical condition (3) then for every
y 1
2(L 2(
S))
N and every
>0, there existsv2L 2 0(;
0) such that jy(v)
=S ;y 1 j (L 2 (S)) N :
Proof. We only sketch the proof here as we use the same method as in Section 2 and for simplicity we will takef = 0.
Given 0
2(L 2(
S))
N, there exists a unique couple (
)2V L 2() =< such that 8 < :
;;(a:r)= 0
: S
;r in ,
div= 0
=;= 0 :
(15)
We can show that if ! is an open subset of such that!\S=, then2
(H 2( !)) N and 2H 1(
!), so thatandare \regular" in the neighborhood
of ;0.
Denoting by P
0 the orthogonal projection from ( L 2(; 0)) N onto L 2 0(; 0) we dene J( 0) = 12
Z ;0 jP 0( @ @
;)j 2
d+j
Then J is strictly convex and continuous and again the coercivity of J is
equivalent to the following unique continuation property : Let () be solution of (15) such that
P 0(
@ @
;) = 0 on ; 0
: (16)
Then = 0, = Cst in so that
0 = 0. The main di"culty is then to
prove the unique continuation property with a2(L
1())N only. We have
(by adding a suitable constant to ) 8
<
:
;;(a:r)=;r in S
ext,
div= 0 in S
ext,
=; 0 = 0
( @ @
;) =;
0 = 0 :
From a result by C. Fabre and G. Lebeau (see 2]) which relies on new ad'hoc Carleman inequalities, we know that = 0 in S
ext and therefore
= 0 on S. Now inS
int we have a homogeneous Stokes system with zero right hand
side which shows that = 0 inS
int. Therefore,
= 0 and=Cstin , so
that 0 = 0.
As a consequence, there exists a unique ^ 0
2(L 2(
S))
N such that
J( ^
0) = min
0 2(L
2 (S))
N J(
0) :
If we write (^^) the solution of (15) associated with ^ 0, ^
v=;P 0(
@ ^ @
;^ )
and ^y=y(^v), it is easy to show that jy^
=S ;y
1 j
(L 2
(S)) N
:
Remark 11. It is possible to extend the result to some nonlinear systems
by using a xed point argument, taking a = a(y). But we need a to be
bounded in (L
1())N and to satisfy
diva= 0. So it seems di"cult to
ob-tain relevant nonlinearities. For the time dependent problem, the condition
diva= 0 is no more necessary and we can obtain a wider class of
nonlinear-ities. Unfortunately the case of Navier Stokes equations cannot be treated up to now.
As for the case of Laplace equation, the method leads to numerical algo-rithms. The numerical results will be presented together with a complete version of our work in 9].
References
1] C.Fabre.Uniqueness results for general Stokes equations and their consequences in control problems for linear and nonlinear equations. To appear in ESAIM : Control, Optimisation and Calculus of Variations.
2] C.Fabreand G.Lebeau.Prolongement unique des solutions de l'equation de Stokes. To appear in Comm. in P.D.E.
3] C.Fabre, J.-P.Puel, E.Zuazua.Approximate controllability of the semilinear heat equation. Proc. of the Royal Soc. of Edinburgh, 125A, p. 31-61 (1995).
4] J.-L.Lions.Exact controllability, stabilization and perturbations for distributed sys-tems. SIAM Rev., 30, p. 1-68 (1988).
5] J.-L.Lions.Remarques sur la contr^olabilite approchee. Proceedings of \Jornadas His-pano Francesas sobre control de sistemas distribuidos", University of Malaga, Spain (1990).
6] J.-L.Lions.Remarks on approximate controllability. Israel J. of Math. (1992). 7] J.-L.Lions and E.Magenes.Problemes aux limites non homogenes et applications.
8] A.Ossesand J.-P.Puel.On the controllability of the semilinear Laplace equation on an interior curve. To appear.
9] A.Osses and J.-P.Puel.Contr^olabilite sur une courbe interieure pour l'equation de Stokes stationnaire avec un potentiel. To appear.
10] W.Press, S.Teukolsky, W.Vetterling, B.Flannery.Numerical recipes in C, the art of scientic computing. Cambridge, Cambridge University Press (1992).
11] J.-C.Saut and B.Scheurer. Sur l'unicite du probleme de Cauchy et le prolonge-ment unique des equations elliptiques a coecients non localeprolonge-ment bornes. Journal of Dierential Equations 43, p. 28-43, (1982).
12] R.Temam.Navier Stokes Equations. North Holland, Amsterdam (1977).
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